# Hervé Jacquet

Hervé Jacquet
Born 1939 (age 77–78)
France
Nationality French
Alma mater École Normale Supérieure
Scientific career
Fields Mathematics
Institutions Columbia University

Hervé Jacquet is a French American mathematician, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern number theory.

## Career

Jacquet entered the École Normale Supérieure in 1959 and obtained his doctorat d'état under the direction of Roger Godement in 1967. He held academic positions at the Centre National de la Recherche Scientifique (1963–1969), the Institute for Advanced Study in Princeton (1967–1969), the University of Maryland at College Park (1969–1970), the Graduate Center of the City University of New York (1970–1974), and became a Professor at Columbia University in 1974, becoming Professor Emeritus in 2007.

## Mathematical work

The book by Hervé Jacquet and Robert Langlands on ${\displaystyle GL(2)}$[1] was an eclipsing event in the history of number theory. It presented a representation theory of automorphic forms and their associated L−functions for the general linear group ${\displaystyle GL(2)}$, establishing among other things the Jacquet-Langlands correspondence which explains very precisely how automorphic forms for ${\displaystyle GL(2)}$ relate to those for quaternion algebras. Equally important was the book by Roger Godement and Hervé Jacquet,[2] which defined, for the first time, the standard L-functions attached to automorphic representations of ${\displaystyle GL(n)}$, now called Godement-Jacquet L-functions, and proved their basic, oft-used analytic properties. The papers with Shalika[3][4] and the papers with Piatetski-Shapiro and Shalika[5][6][7] pertain to L-functions of pairs, called the Rankin-Selberg L-functions, attached to representations of ${\displaystyle GL(n)}$ and ${\displaystyle GL(m)}$, and the so-called converse theorem, which are crucial to our understanding of automorphic forms. A basic ingredient of this effort was an elaboration of properties of Whittaker models and functions, which Jacquet had made contributions to since his thesis. The papers with Shalika also established the uniqueness of isobaric decompositions of automorphic forms on ${\displaystyle GL(n)}$, thus providing evidence for certain conjectures of Langlands. In the mid-eighties, Jacquet forayed into a new territory in the field and created[8][9][10] the relative trace formula in representation theory, an important tool in modern number theory, which vastly generalizes the Kuznetsov and Petersson formulae from the classical setup. While the usual Selberg trace formula, as well as its generalizations due to Arthur, consists in developing an expression for the integral of the kernel over the diagonal, the relative version integrates the kernel over other appropriate subgroups.

## Awards and honors

He was elected corresponding member of the Académie des Sciences in 1980. In 2012 he became a fellow of the American Mathematical Society.[11] He was elected to the American Academy of Arts and Sciences in 2013.[12]

## References

1. ^ H. Jacquet and R. P. Langlands. Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol.114, Springer-Verlag, Berlin (1970).
2. ^ H. Jacquet and J. A. Shalika. A non-vanishing theorem for zeta functions of GLn. Invent. Math., 38(1):1–16, 1976/77.
3. ^ H. Jacquet and J. A. Shalika. On Euler products and the classification of automorphic forms, I, Amer. J. Math. 103(3): 499–558 (1981).
4. ^ H. Jacquet and J. A. Shalika. On Euler products and the classification of automorphic forms, II, Amer. J. Math. 103(4): 777–815 (1981).
5. ^ H. Jacquet, I. I. Piatetski-Shapiro and J. A. Shalika. Automorphic forms on GL(3). I. Ann. of Math. (2), 109(1):169–212, 1979.
6. ^ H. Jacquet, I. I. Piatetski-Shapiro and J. A. Shalika. Automorphic forms on GL(3). II. Ann. of Math. (2), 109(2):213–258, 1979.
7. ^ H. Jacquet, I. I. Piatetski-Shapiro and J. A. Shalika. Rankin-Selberg convolutions. Amer. J. Math., 105(2):367–464, 1983.
8. ^ H. Jacquet. Sur un résultat de Waldspurger. Ann. Sci. École Norm. Sup. (4), 19(2):185–229, 1986.
9. ^ H. Jacquet. Représentations distinguées pour le groupe orthogonal. C. R. Acad. Sci. Paris Sér. I Math., 312(13):957–961, 1991.
10. ^ H. Jacquet and K. F. Lai. A relative trace formula, Compositio Mathematica, 54(2), 243–310 (1985).
11. ^ List of Fellows of the American Mathematical Society, retrieved 2013-01-26.
12. ^ Newly elected members, American Academy of Arts and Sciences, April 2013, retrieved 2013-04-24.